Question

             Lenny's, a national restaurant chain, conducted a study of the factors affecting demand (sales). The

             following variables were defined and measured for a random sample of 30 of its restaurants:

Y	= Annual restaurant sales ($000)
X1	= Disposable personal income (per capita) of residents within 5 mile radius
X2	= License to sell beer/wine (0 = No, 1 = Yes)
X3	= Location (within one-half mile of interstate highway--0 = No, 1 = Yes)
X4	= Population (within 5 mile radius)
X5	= Number of competing restaurants within 2 mile radius

The data were entered into a computerized regression program and the following results were obtained:

MULTIPLE R	.889
R-SQUARE	.79
STD. ERROR OF EST.	.40

ANALYSIS OF VARIANCE
	DF	Sum Squares	Mean Sqr.	F-Stat
Regression	5	326.13	65.226	18.17
Error	24	86.17	3.590
Total	29	412.30

Variable	Coefficient	Std. Error	T-Value
Constant	0 .363	0 .196	1.852
X-1	0 .00275	0 .00104	2.644
X-2	76.65	93.70	0 .818
X-3	164.3	235.4	0 .698
X-4	0 .00331	0 .00126	2.627
X-5	46.2	12.1	-3.818

Questions:

(a)	Give the regression equation for predicting restaurant sales.
(b)	Give an interpretation of each of the estimated regression coefficients.
(c)	Which of the independent variables (if any) are statistically significant at the .05 level in "explaining" restaurant sales (assume the critical T-Value is 2)?
(d)	What proportion of the variation in restaurant sales is "explained" by the regression equation?
(e)	Perform an F-test (at the .05 significance level) of the overall explanatory power of the regression model(assume that the critical value of F is =.2.62).

Answer 1

a) The regression equation would look like:

Y = 0 + 1X1 + 2X2 + 3X3 + 4X4 + 5X5 + i

Thus, Y = 0.363 + 0.00275X1 + 76.65X2 + 164.3X3 + 0.00331X4 + 46.2X5

b) A 1 unit Increase in PDI(per capita) of residents within 5 mile radius leads to a 0.00275 unit Increase in sales. A license to sell bear/wine Increases the sales by 76.65 units. A location within one-half mile of interstate highway Increase the sales by 164.3 units. An increase in population (within 5 mile radius) by 1 unit leads to a 0.00331 unit Increase in sales. Finally, an increase in 1 competing restaurant within 2 mile radius increases the sale by 46.2 units.

c) Only X1, X4 and X5 are statistically significant at the given level of significance since their |tstat|>crit. Rest all have tstat<crit, therefore insignificant.

d) Since R^2 is 0.79, therefore about 79% of the variation is explained by the given variables on the dependent variable.

e) Fstat of regression is given as 18.17 which is greater than Fcrit of 2.62, therefore the model is overall significant.